his5 - Hilbert Space Explorer

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Theorem his5 28071

Description: Associative law for inner product. Lemma 3.1(S5) of [Beran] p. 95. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)

**Assertion**
Ref	Expression
his5	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ih (𝐴 ·_ℎ 𝐶)) = ((∗‘𝐴) · (𝐵 ·_ih 𝐶)))

**Proof of Theorem his5**
Step	Hyp	Ref	Expression
1		hvmulcl 27998	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐶) ∈ ℋ)
2		ax-his1 28067	. . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ (𝐴 ·_ℎ 𝐶) ∈ ℋ) → (𝐵 ·_ih (𝐴 ·_ℎ 𝐶)) = (∗‘((𝐴 ·_ℎ 𝐶) ·_ih 𝐵)))
3	1, 2	sylan2 490	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ (𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ)) → (𝐵 ·_ih (𝐴 ·_ℎ 𝐶)) = (∗‘((𝐴 ·_ℎ 𝐶) ·_ih 𝐵)))
4	3	3impb 1279	. . 3 ⊢ ((𝐵 ∈ ℋ ∧ 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ih (𝐴 ·_ℎ 𝐶)) = (∗‘((𝐴 ·_ℎ 𝐶) ·_ih 𝐵)))
5	4	3com12 1288	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ih (𝐴 ·_ℎ 𝐶)) = (∗‘((𝐴 ·_ℎ 𝐶) ·_ih 𝐵)))
6		ax-his3 28069	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·_ℎ 𝐶) ·_ih 𝐵) = (𝐴 · (𝐶 ·_ih 𝐵)))
7	6	3com23 1291	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ℎ 𝐶) ·_ih 𝐵) = (𝐴 · (𝐶 ·_ih 𝐵)))
8	7	fveq2d 6233	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (∗‘((𝐴 ·_ℎ 𝐶) ·_ih 𝐵)) = (∗‘(𝐴 · (𝐶 ·_ih 𝐵))))
9		hicl 28065	. . . . . 6 ⊢ ((𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐶 ·_ih 𝐵) ∈ ℂ)
10		cjmul 13926	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝐶 ·_ih 𝐵) ∈ ℂ) → (∗‘(𝐴 · (𝐶 ·_ih 𝐵))) = ((∗‘𝐴) · (∗‘(𝐶 ·_ih 𝐵))))
11	9, 10	sylan2 490	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ)) → (∗‘(𝐴 · (𝐶 ·_ih 𝐵))) = ((∗‘𝐴) · (∗‘(𝐶 ·_ih 𝐵))))
12	11	3impb 1279	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∗‘(𝐴 · (𝐶 ·_ih 𝐵))) = ((∗‘𝐴) · (∗‘(𝐶 ·_ih 𝐵))))
13	12	3com23 1291	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (∗‘(𝐴 · (𝐶 ·_ih 𝐵))) = ((∗‘𝐴) · (∗‘(𝐶 ·_ih 𝐵))))
14		ax-his1 28067	. . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ih 𝐶) = (∗‘(𝐶 ·_ih 𝐵)))
15	14	3adant1 1099	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ih 𝐶) = (∗‘(𝐶 ·_ih 𝐵)))
16	15	oveq2d 6706	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((∗‘𝐴) · (𝐵 ·_ih 𝐶)) = ((∗‘𝐴) · (∗‘(𝐶 ·_ih 𝐵))))
17	13, 16	eqtr4d 2688	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (∗‘(𝐴 · (𝐶 ·_ih 𝐵))) = ((∗‘𝐴) · (𝐵 ·_ih 𝐶)))
18	5, 8, 17	3eqtrd 2689	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ih (𝐴 ·_ℎ 𝐶)) = ((∗‘𝐴) · (𝐵 ·_ih 𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ‘cfv 5926 (class class class)co 6690 ℂcc 9972 · cmul 9979 ∗ccj 13880 ℋchil 27904 ·_ℎ csm 27906 ·_ih csp 27907

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-hfvmul 27990 ax-hfi 28064 ax-his1 28067 ax-his3 28069

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-po 5064 df-so 5065 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-2 11117 df-cj 13883 df-re 13884 df-im 13885

This theorem is referenced by: his52 28072 his35 28073 normlem0 28094 normlem9 28103 bcseqi 28105 polid2i 28142 pjhthlem1 28378 eigrei 28821 eigposi 28823 eigorthi 28824 brafnmul 28938 lnopunilem1 28997 hmopm 29008 cnlnadjlem6 29059 adjlnop 29073

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